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many workers, including Riemann, Christoffel, Ricci, Levi-
Civita, Lie, and Einstein - all names well known to readers of
popular accounts of relativity; the whole vast programme was
originated by the early workers hi the theory of algebraic
invariants, of which Cayley and Sylvester were the true
, As a second example, imagine a knot to be looped in a string
whose ends are then tied together. Pulling at the knot, and
running it along the string, we distort it into any number of
'shapes'. What remains 'invariant', what is 'conserved', under
all these distortions which, in this case, are our transformations?
Obviously neither the shape nor the size of the knot is invariant.
But the 'style* of the knot itself is invariant; in a sense that
need not be elaborated, it is the same sort of a knot whatever we
do to the string provided we do not untie its ends. Again, in the
older physics, energy was 'conserved'; the total amount of
energy in the universe was assumed to be an invariant, the same
under'all transformations from one form, such as electrical
energy, into others, such as heat and light.
Our third illustration of invariance need be little more than
an allusion to physical science. An observer fixes his ^position*
in space and time with reference to three mutually perpendi-
cular axes and a standard timepiece. Another observer, moving
relatively to the first, wishes to describe the same physical event
that the first describes. He also has his space-time reference
system; his movement relatively to the first observer can be
expressed as a transformation of his own co-ordinates (or of the
other observer's). The descriptions given by the two may or
may not differ in mathematical form, according to the parti-
cular kind of transformation concerned. If their descriptions do
differ, the difference is not, obviously, inherent La the physical
event they are both observing, but in their reference systems
and the transformation. The problem then arises to formulate
only those mathematical expressions of natural phenomena
which shall be independent, mathematically, of any particular
reference system and therefore be expressed by all observers in
the same form. This is equivalent to finding the invariants of
the transformation which expresses the most general shift in